In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking the tensor product of this with any module $N$ gives an exact sequence $0 \rightarrow M' \otimes N \rightarrow M \otimes N \rightarrow M'' \otimes N \rightarrow 0$.
Later in the book, in Exercise 5.2.8, page 193, he suggests using this Proposition to show that given a long exact sequence which is eventually $0$:
$$ 0 \rightarrow M_0 \rightarrow M_1 \rightarrow \cdots $$
of flat modules, taking the tensor product with an arbitrary module still gives an exact sequence
$$ 0 \rightarrow M_0 \otimes N \rightarrow M_1 \otimes N \rightarrow \cdots$$
How does this generalization follow? It appears that in the proof of the Proposition, he doesn't really use the surjectivity of $M \rightarrow M''$, but then again, all he proves is that $M' \otimes N \rightarrow M'' \otimes N$ is injective; the full exactness being something that always happens for tensor products of short exact sequences.